I've posted all 787,444 solutions for $N=7$ at [this link](https://home.gwu.edu/~maxal/N7.txt). 

To compute these solutions, I've used bounds similar to those proposed in the answer by David desJardins, but only for terms $x_1, \dots, x_{N-2}$. The remaining two terms satisfy the equation:
$$\frac{a}{c}+\frac{1}{x_{N-1}}+\frac{1}{x_N} = \frac{b}{c} \big(1-\frac1{x_{N-1}}\big)\big(1-\frac1{x_N}\big),$$
where $\frac{a}{c} := \sum_{i=1}^{N-2} \frac1{x_i}$, $\frac{b}{c} := \prod_{i=1}^{N-2} (1-\frac1{x_i})$, and $\gcd(a,b,c)=1$, which is equivalent to
$$\big((a-b) x_{N-1} + b+c\big)\cdot \big((a-b) x_N + b+c\big) = b(a-b) + (b+c)^2$$
from where the suitable values of $x_{N-1}$ and $x_N$ can be determined efficiently by factoring of the right hand side.

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I've also added solution counts to the OEIS as [sequence A369469](https://oeis.org/A369469). A similar [sequence A369470](https://oeis.org/A369470) enumerates solutions with possibly equal terms.